Notation. When \(\mathcal {F}\)  is the family of finite groups, we use indistinctly the notations \(\underline{E}G\)  or \(E_{\mathcal {F}}G\) , and similarly when \(\mathcal {F}\)  is the family of virtually cyclic groups and \({\underline{\underline{E}}}G\)  and \(E_{\mathcal {F}}G\)  notations for the corresponding classifying space. If \(\mathcal {F}\)  is the family that only contains the trivial group, the superindex in the Bredon homology will be supressed, as it is ordinary homology in this case.
It is worth noticing that there is an algebraic definition of \(H_*^{\mathcal {F}}(G,M)\) , however there is an isomorphism \(H_*^{\mathcal {F}}(G,M)\simeq H_*^{\mathcal {F}}(E_{\mathcal {F}}G,M)\)  between the algebraic and the topological definitions of Bredon homology [1]}.
To not overload the paper with unnecessary notation, we commonly denote this homology groups by \(H_*^{\mathcal {F}}(G,M)\) , although as said above, we mainly deal with the topological definition.
